Consider $(\mathbb{R}^n, +)$, endowed with standard metric. Let $H$ be its subgroup, with the property that there is a neighbourhood of $0$, $V$, such that $V \cap H = \{0\}$. How do I prove that $H \cong \mathbb{Z}^k$, $k \le n$?
I am more proficient with analysis than algebra and my intuition is both have to be used here but am currently at loss how to do this.